Quadratic Equations in Square Roots


For solving Quadratic Equations in Square Roots, it is required to square both sides entirely, but not individually. For instance,
$$ (1 + 2)^2 = 3^2 \\
1^2 + 2^2 \ne 3^2 $$
It is important to check the solutions to see they work for the original equation, if the original equation is squared.

Worked Examples of Quadratic Equations in Square Roots

Solve \( \sqrt{x+5} + \sqrt{x-2} = \sqrt{5x-6} \) .

\( \begin{aligned} \require{color} \displaystyle
\Big(\sqrt{x+5} + \sqrt{x-2}\Big)^2 &= \sqrt{5x-6}^2 &\color{red} \text{square both sides} \\
x+5 + 2 \sqrt{x+5} \sqrt{x-2} + x-2 &= 5x-6 \\
2 \sqrt{x+5} \sqrt{x-2} &= 3x-9 \\
4(x+5)(x-2) &= (3x-9)^2 &\color{red} \text{square both sides} \\
4(x^2 + 3x – 10) &= 9x^2 – 54x + 81 \\
5x^2 -66x + 121 &= 0 \\
(5x-11)(x-11) &=0 \\
x = \frac{11}{5} \text{ or } 11 \\
\end{aligned} \\ \)
\(
\text{Check whether the answers are OK, but } x= \displaystyle \frac{11}{5} \text{ does not make the original equation.} \\
\text{So the answer is only } x=11. \\
\)

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